Before
we even began to define the vector space we need to introduce some notation
which will use later on. So we introduce these couple of notations:
1) R
and C will represent the set of real and complex numbers respectively. Real and
complex numbers will be called real and complex scalars. The set of all
integers will be called Z, and set of natural numbers will be N,
2) The
organized n-tuple of scalars is every series of n-tuple scalars. The organized
n-tuple of scalars will be written with a pair of brackets. For example if
x1,x2,…,xn given scalars, then the organized n-tuple of scalars will be written
in the following form: (x1,x2,…,xn) and this will represent the organized
n-tuple of those scalars. x1 is in first
position, x2 is on a second position ect. The set of n-tuple scalars
(x2,x1,…,xn) isn’t equal to the previous organized n-tuple of scalars.
3) With
Vn or Rn we will label the set of all organized n-tuple real numbers. Word
“All” represents all of n-tuple, each of which is in the form x=(x1,x2,…,xn),wherein
each xi is arbitrary real number. If we write x,y,z e Vn that means that x,y
and z are organized n-tuple of real numbers. Their components are xi,yi and zi
or: X=(x1,x2,…,xn),y=(y1,y2,…,yn) and z=(z1,z2,…,zn).
4) N-tuple
of complex numbers will be written with small Latin letters, but we will
indicate that their components are complex numbers. The set of all n-tuple will
be labeled with Cn. For example Z=(z1,z2,…,zn) and W=(w1,w2,…,wn) are from Cn,
if zi and wi are elements of complex set Cn for every 1<i<n.
5) The
general vector spaces will also be denoted with large Latin letters, and their
elements (vectors) will be denoted with small Latin letters. Sets and subsets
will be denoted with large Latin or calligraphic letters. For example S or A.
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